Lec 7-8

Question 1

What are examples of domains in which we deal with dynamical systems with uncertainty?

Answer 1

Natural language analysis, mathematical finance, disease development

Question 2

How can we describe dynamical systems with uncertainty?

Answer 2

Stochastic processes: Uncertainty model for system that changes over time.

A sequence or random variables X_N: Probability space to outcome.

It can be interpreted as a Bayesian network (with infinitely many vertices)

Question 3

What is the problem with normal stochastic processes?

Answer 3

To paramaterise up to time n we would need the initial mass function p(x1) and the conditional distributions for all k, where the number of parameters grows exponentially with n

Question 4

What is the Markov property?

Answer 4

MCs assume that for every timepoint k the next timepoint k is independent of k (everything before the node before it).
P(X_(K+1)|X_K)

Question 5

What are these conditional pmts called?

Answer 5

Transition probabilities

Question 6

What does px mean in case of discrete vs continuous RV’s?

Answer 6

For discrete: A probability mass function
For continuous: A probability density function

Question 7

What is Monte Carlo Estimation?

Answer 7

The process of estimating the Expected Value by sampling px

Question 8

What is Inverse-Transform Sampling?

Answer 8

If we have a RV and a known CDF mapping from 0 to 1. Sample x from px:
- Find the inverse CDS or quantile function
- Sample u uniformly at random from [0,1]

Question 9

What are practicalities with the Inverse-Transform sampling of RV?

Answer 9

It requires knowing F-1. Often but not always available. If not available, more sophisticated methods are necessary.